Black Holes and Dimensional Analysis - Sixty Symbols

Featuring Professor Ed Copeland with a look at dimensional analysis and how it can be used on black holes, among other things... More links and info below ↓ ↓ ↓


This project features scientists from The University of Nottingham


Sixty Symbols videos by Brady Haran
Animation by Pete McPartlan

[ASparkyB]

Professor Copeland's handwriting certainly looks like the handwriting of someone who has been writing on graph paper for their entire life. I love it.

[kalleguld]

Dimensional analysis is by far the most useful thing I learned in physics class. I learned a lot of useful things, but this is the best.

[eumoria]

Professor Copeland is an amazing guy... I was never in physics but as an educator he's fantastic.

[feandil]

I remember my physics teacher had a custom made red stamp: "non dimensional", she would stamp our papers and wouldn't bother to check anything else if the result was non dimensional. It taught us some really good lesson, because if you get the units right the rest is pretty easy.

[julian]

Videos with Professor Copeland are certainly a treat! Love it!

[donaldasayers]

Dimensional analysis was not on the A-level syllabus when I was a kid, but we were taught it as a means of checking answers in exams to make sure you were in the right ballpark. I have found it very useful, but care has to be taken when things start rotating, like torque.

[ZeedijkMike]

Professor Copeland is such a pleasure to listen to and watch. Blings a smile to my face - and I'm learning at the same time.

[ragnkja]

Pluto is a fine example of an object in orbit around the Sun.

[maxtrax]

You should make more videos with Ed Copeland. I truly like him.

[cordial]

I love how you guys are just having a laugh together while Ed explains his point.

[davidalexallen]

Love Professor Copeland's videos, he always explains things so well

[SteveGuidi]

I'll never forget the day I was first introduced to Dimensional Analysis from my calculus professor. The lecture started with analyzing the units of Newton's Laws, as Professor Copeland demonstrated in this video. Then after a few steps my professor exclaims something along the lines of "there you have it, Kepler's Laws of motion without doing any hard math whatsoever." What an epiphany that was for me as a young student!

[DrumsTheWord]

I love Prof Copeland. What a great chap and educator!

[duggydo]

Ed is hands down the best presenter on any of Brady's channels.

[pinkdispatcher]

I found it fascinating that one of the standard works on Aerodynamics, "Fundamentals of Flight" by Richard Shevell, also uses dimensional analysis as a tool to come up with how the different properties of air, geometry and motion influence the result.

[webspiderc]

I like Professor's representation so much. He is always calm and shows those complex things in a simple way to be easily understood.

[vincentpelletier]

This is also useful when doing more complex calculations, when carrying around these G and c constants during long derivations can be cumbersome, so when working in General Relativity we can set the units such that c = 1 and G = 1. Essentially, time, mass and distance are all measured in the same units, say meters. To get time, do dimensional analysis: you have time is X meters, you want it in seconds. c has units of meters / seconds, so you divide X by c and get the number of seconds.

Thus, the Schwarzchild radius in this case can be written as R_S = 2 M. To get everything in normal units, figure out how many factors of G and c you need to get meters on one side, kilograms on the other. So R_S = 2 G M / c^2

Because of this, I can always remember the mass of the Sun, it is 1.5km (that is, it's Schwarzchild radius is 3km), and easy number to remember. How many kilograms is that? I leave that as an exercise, I am not going to remember *that*.

[dragonfly.effect]

My favorite bit of dimensional analysis related to black holes is this:

Many people (unfortunately including physicists who should know better) like to say things like "BHs are the densest things in the universe, " or "a mass becomes a BH when it shrinks to an extreme density".  Some of this confusion may be conflating the density of the BH with the theoretically infinite density of the (hypothetical) central singularity.  But consider the following:
1.  Rₛ = 2GM/c²; but 2G/c² is a (universal) constant, so
2.  Rₛ is proportional to M.
3.  On the other hand, density ρ = M/V (where V = volume), so
4.  (ignoring constants) ρ is proportional to M/Rₛ³,
5.  which in turn (by point 2) is proportional to M/M³ = 1/M².

i.e., the bigger the BH (measured by mass or by radius), the sparser it is.
e.g., the density of a solar-mass BH would be ≈ 20 trillion g/cc, but the density of a 10 million solar mass ("supermassive") BH would be more like 0.2 g/cc, or about ⅕ that of water.  (Etc.)

[SudaNIm]

I just absolutely love Prof. Copeland, his episodes are always my favorite! Most everyone Brady features are both brilliant and fascinating; Prof. Copeland is of course no exception but his palpable enthusiasm and sincere humility set him apart. He’s just effortlessly engaging, for me at least and has the character of the ideal educator. 🤓

[harper]

always appreciate more of prof copeland's commentary